Locally conformally Kähler manifolds and their cohomology

A Hermitian metric on a complex manifold is locally conformally Kähler (LCK) if it is conformal to a Kähler metric around each point.Many of the complex non-Kähler manifolds admit such a metric. In this talk, I will discuss several examples, focusing on complex surfaces.

LCK manifolds provide a natural context for the study of twisted cohomology, also known as Morse-Novikov, which is the cohomology with values in a flat bundle. We compute it for Inoue surfaces and explain the relation between the twisted cohomology and their LCK geometry.